Splitting, merging, and the stability of subject boundaries

20.1 Two cores and a controlled merger

Consider two bits initially evolving by separate noisy persistence channels, with retention contrasts $a=4/5$ and $b=1/2$. Their internal dependence graph has two self-loops and no cross edge. Each one-tick physical return has positive TV, and the native records are the actual next register values. Both candidates qualify, so A1 assigns two perspectives.

For $g\in(0,1]$, install one native stochastic mechanism with the joint update

where $T_0$ is the independent persistence update and $T_1$ is the cross-update of Equation 19.1. The mechanism, with both of its internal input ports, is executed at every tick. Its native route certificate records two applications of that installed mechanism and the tour $A\to B\to A$. The convex mixture is a representation of its stochastic law, not a schedule that physically omits the mechanism on some branches. A different implementation that actually switches between incompatible topologies would require its own certificate.

The one-step cross-dependence is nonzero for every $g>0$. Using centered binary coordinates, the two-step mean response of $A$ to its own earlier value has coefficient

The shared operating contract includes that two-tick return template. The route certificate verifies execution of the two installed coupling applications on every outcome branch; it does not assert successful transmission of a distinction along every noisy sample path. Exact native next-state records supply nontrivial prediction and automatic predictive-fibre congruence. There is therefore one qualifying combined core under these stated mechanism and route assumptions. A3 ends the two predecessor episodes at the merger and begins a successor. Removing the cross mechanism restores two cores and starts their successor episodes.

If communication is instead through ports typed as external, the graph used for A1 omits the external return route and retains the two internal cores. The physical communication is not denied; it is classified as interaction between the nominated vessels. The different result makes boundary typing load-bearing. A universal theory would need to justify that typing from deeper physical structure rather than select it according to an intuitive subject count.

20.2 Arbitrarily weak coupling gives a discontinuity

The exact nonzero-edge rule is not generally robust. Let $T_g=(1-g)T_0+gT_1$, with $T_0$ the independent update and $T_1$ a bidirectionally coupled update. Then

For the same initial preparation and $H$ steps, an auxiliary mixture coupling gives

With those compatible return certificates, the subject count changes from two at $g=0$ to one for every positive $g$. Thus arbitrarily small finite-horizon physical changes can produce a discontinuous subject assignment.

This is not a formal contradiction. A theory can contain a sharp threshold. It is, however, a substantive prediction and a robustness problem. An $\epsilon$-effective graph that ignores effects below a fixed discrimination scale gives a more stable operational classification, but it is an added resolution convention, not the exact law secretly preserved. Inference about actual subject boundaries should report this sensitivity rather than hide it behind the word “integration.”

If every estimated edge strength is separated from a declared threshold by a margin larger than the error bound, the thresholded graph and its SCC partition are stable. This follows because no edge can cross the threshold under the allowed perturbation. Near zero in the exact rule there need be no such margin. Physical model uncertainty and psychophysical discontinuity must therefore be analyzed together.

20.3 Copying, reset, and continuation

Let a donor and receiver be distinct lineaged carriers. A reversible copy transfers a selected record, not the donor's physical lineage. If each belongs to a separate qualifying core, A3 does not identify their episodes merely because their memory contents agree. The same applies to an artificial-system handoff that reproduces a relational task state in another instance.

A local SWAP reset can preserve the core's physical lineage while changing its working memory. Whether qualification persists depends on the actual post-reset operation. A1 may continue to admit the core; A2 generally changes the pointed content if future internal laws change. Resetting a narrative label does not automatically reset a subject, and preserving a narrative label does not automatically preserve subject identity through a genuine physical replacement.

A narrative variable can belong to the core's retained organization, to an external record, or to a report channel. Its causal location matters. Changing a self-description that participates in endogenous continuation can change the assigned phenomenal structure; changing only an external label need not. The philosophical statement that ego is not awareness therefore leaves room for profound experiential consequences of the personal model without treating every statement about identity as a change of subject. A core can continue while its self-description changes, and a similar self-description can occur in a distinct successor episode.

20.4 Finite partitions and universal unity

Communication by itself does not establish subject unity. Two machines, two brains or a loose network are not one subject merely because they exchange signals. A brain and body, or a distributed process, can receive a unified assignment only if the relevant whole is one justified realization with the internally typed organization and return tests required by A1. A tiny exchange outside that organization does not automatically merge subjects.

This qualification does not remove the exact weak-coupling consequence above. Once a positive coupling belongs to the certified internal organization and supports the required joint return, an arbitrarily weak exchange can merge the cores under the present law. Boundary typing must be justified independently; it cannot be chosen after the fact to preserve a preferred count.

The construction determines a subject partition only relative to the nominated realization. It does not infer a universal observer or a numerical One from $\unsplit$. Nor does it infer independent subjects for every mathematical subsystem. The distinction between a physical component, an operational assembly, a qualifying core, a phenomenal perspective, and a narrative person must be retained.

This is where the philosophical attraction and cost of the proposal meet. A0 permits a common ontological ground without demanding separate mental substances. A1 supplies a local individuation rule. A2 determines relational content. A3 chooses a temporal identity doctrine. The combination is coherent and evaluable in a finite model, but each additional identification creates an empirical or philosophical responsibility. The theory's seriousness depends on accepting those responsibilities rather than claiming that graph theory has answered them without assumptions.